Abstract
The existence of families of bounded (with respect to space variables) solutions of the multidimensional nonlinear Schrödinger equation is established and also their asymptotic properties are studied. The investigation consists of two steps. At first, the initial equation is reduced with the help of ansatzes of special form to a system of ordinary differential equations, and then qualitative analysis of each equation of this system is carried out.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
I. O. Parasyuk and V. I. Fushchich, “Qualitative analysis of families of bounded solutions of the three-dimensional nonlinear Schrödinger equation,” Ukr. Mat. Zh.,42, No. 10, 1344–1350 (1990).
V. I. Fushchich [W. I. Fushchich] and N. I. Serov, “On some exact solutions of three-dimensional nonlinear Schrödinger equation,” J. Phys. A, Math. Gen.,20, 929–933 (1987).
V. I. Fushchich, V. M. Shtelen', and N. I. Serov, Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics [in Russian], Naukova Dumka, Kiev (1989).
R. Bellman, Stability Theory of Differential Equations, McGraw Hill, New York (1953).
A. A. Samarskii, V. A. Galaktinov, S. P. Kurdyumov, and A. P. Mikhailov, Peaking Modes in Problems for Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1987).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 821–828, June, 1991.
Rights and permissions
About this article
Cite this article
Fushchich, V.I., Parasyuk, I.O. Qualitative analysis of families of bounded solutions of the multidimensional nonlinear Schrödinger equation. Ukr Math J 43, 768–775 (1991). https://doi.org/10.1007/BF01058945
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01058945